A Variational Approach for the Design of the Spatial Four-Bar Mechanism

Transcription

1 Zou, H.L., and Abdel-Malek, K., and Wang, J.Y., (1997), A Varatonal Approach for the Degn of the Spatal Four-Bar Mechanm. Mechanc of Structure of Machne, Vol. 5, No. 1, pp A Varatonal Approach for the Degn of the Spatal Four-Bar Mechanm H. L. Zou K. A. Abdel-Malek J. Y. Wang Center for Computer Aded Degn and Department of Mechancal Engneerng The Unverty of Iowa Iowa Cty, IA 54 ABSTRACT An analytcal formulaton for computng knematc entvty of the patal four-bar mechanm preented. The expermental code developed ued to compute an aembled confguraton for the mechanm due to a degn varaton. The mechanm modeled ung graph theory where a body defned a a node and a knematc ont defned a an edge. The phercal ont cut to convert the model nto a tree tructure by cuttng an edge and ntroducng contrant. The effect of varaton n mechanm degn ung concept of vrtual dplacement and rotaton ntroduced. The varaton of the phercal contrant computed whle mantanng ont-attachment vector and orentaton matrce a varable. A ytem of equaton that ha more degn varable than equaton then olved ung the modfed Moore-Penroe peudo nvere. A recurve formulaton ntroduced to obtan the tate varaton of a body n term of the tate varaton of a uncton body and of the relatve coordnate along the chan. The Jacoban matrx then tranformed from Cartean coordnate pace to ont coordnate pace ung velocty tranformaton matrce. Knematc entvty analye due to changng a ontattachment vector and an orentaton are preented. I. INTRODUCTION The planar and patal four-bar mechanm are ome of the mot ued machne ubytem. Becaue the concept of thee mechanm old and wdely ued, they have been extenvely tuded. In recent year, the empha ha been on the ynthe ung numercal technque. For example, the fnte poton ynthe of a phercal four-bar mechanm ung a mappng approach of the dered poton to pont n an mage pace wa preented by Boddulur and McCarthy [1]. A oluton of the nne-pont path ynthe of four-bar lnkage wa preented by Wampler, et al. [] where the problem formulated and olved ung a combnaton of clacal elmnaton, multhomogeneou varable, and numercal contnuaton method. Another method that ue the contnuaton method n the analy of four-bar lnkage wa reported by Subban and Flugrad [3]. Graphcal ynthe of four-bar lnkage were reported by Norton, et al. [4] and Khanua, et al. [5]. Other ynthe method are reported by Hwang and Chang [6]. 1

2 Computer-aded degn technque were alo ued to automate the nfnte pont coupler curve ynthe ung unform perodc B-plne [7]. The geometry and dynamc of the confguraton pace of the four-bar mechanm ung ngularty theory wa tuded by Yang and Krhnapraad [8]. Work that have dealt wth the tranmon angle of the four-bar mechanm n term of the mechanm moblty analy were preented by Kazerounan and Soleck [9]. The mechanm contant force tranmon charactertc were tuded by Soylemez [1]. The nput-output equaton of the underlyng mechanm wa analytcally tuded for contant-branch moton by Lu and Angele [11] who alo tuded the leat quare optmzaton of planar and phercal mechanm under moblty contrant [1]. A mlar analy wa appled to the mechanm path generator [13]. Sheth and Ucker [14] mplemented graph theory to analyze the topology of multbody ytem n term of relatve coordnate. Graph theory wa alo ued by Wttenburg [15] to handle cloed loop ytem by cuttng ont to form a pannng tree. Wttenburg and Wolz [16] alo preented a cut-body method n a computer program for artculated multbody dynamc. The recurve formulaton ued n th paper wa adapted from the work of Bae and Haug [17, 18] who ued Wttenburg approach to modelng of multbody ytem. Th type of formulaton wa mplemented ung parallel computatonal technque by Hwang, et al. [19] and Hwang and Haug []. II. CONCEPT OF VARIATION IN MECHANISM COMPONENT DESIGN USING VIRTUAL DISPLACEMENTS AND ROTATIONS A patal four-bar mechanm hown n Fg. 1 modeled a a four-body ytem connected by a phercal, a unveral, and two revolute ont. The ground choen a the bae body where t body-fxed reference frame concde wth the global reference frame. Sphercal ont Unveral ont z 4 x 1 Revolute ont x 3 r 1 y 4 z 3 y 3 r 3 4 y 1 z 1 Revolute ont x 4 Fg. 1 Spatal four-bar mechanm

3 The pannng tree of the ytem hown n Fg. where the phercal ont ha been cut. Cuttng the phercal ont reduce the maxmum number of relatve coordnate (three) and ntroduce a mnmum number of cut-ont contrant. Two relatve coordnate q 1 (between the ground and lnk 1) and q 4 (between lnk 3 and the ground) are ued for the two revolute ont. Two coordnate q and q 3 repreent the unveral ont connectng lnk 1 and lnk. A bodyfxed frame xyz defned at the ont connected to the nboard body of the knematc chan, and the coordnate ytem xyz are defned at ont that are connected to the outboard bode. Lnk Unveral ont cut ont Lnk 1 1 revolute ont revolute ont Lnk 3 4 Ground Fg. (a) Graphc repreentaton of the patal four-bar mechanm In order to develop the formulaton for the cut phercal ont, t convenent to wrte the phercal contrant a a requrement that necetate a par of pont on two bode to concde. A neceary and uffcent condton for O and O to concde [1] that d uch that Φ ph ( O, O ) r r (1) where Φ ph denote a vector of contrant functon for the phercal ont. The three calar contrant equaton of the phercal ont are defned a Φ ph r3 3 r 3 () Snce the patal four-bar mechanm ha four generalzed coordnate and three contrant equaton, t ha one degree-of-freedom. In dervng the varatonal contrant, t neceary to keep the ont-attachment vector and the orentaton matrce a varable n order to later obtan ther knematc entvty. III. VARIATION IN MECHANISM COMPONENT DESIGN USING VIRTUAL DISPLACEMENT AND ROTATION In th ecton, the concept of vrtual dplacement and rotaton wll be ntroduced. In order to account for the effect of degn varaton, t neceary to develop a formulaton for the contrant varaton whle mantanng varable parameter. The vrtual dplacement and rotaton vector (ont-attachment vector and orentaton matrce) wll be mantaned a varable n order to later obtan ther entvty. The tudy of the effect of degn change on knematc varable and the tudy of degn change requred to preerve knematc confguraton wll be addreed and demontrated va numercal example. 3 3

4 Snce the phercal contrant of the four-bar mechanm cut, conder t varaton a δφ ph δr δ δr δ (3) where ont-attachment vector can be reolved n the body reference frame a A (4) where A a tranformaton matrx from the body coordnate ytem xyz to the global coordnate ytem xyz. Takng the varaton of both de of Eq. (4) yeld δ δa Aδ (5) Note that δ and δ are varable. The vrtual rotaton matrx δ π ~ (a kew-ymmetrc matrx) wa defned by Ta and Haug [] a δ~π δaa T (6) where the tlde operator (.e., the ymbol ~) ued to denote a kew-ymmetrc matrx generated by the aocated vector. Smlar to the vrtual rotaton matrx δ π ~ aocated wth the matrx A, defne a vrtual rotaton matrx δ ~ ξ aocated wth the tranformaton matrx C uch that ~ δ T ξ δcc (7) where C the tranformaton matrx from ont reference frame x y z to body reference frame xyz. To mplfy the computaton of vrtual rotaton, Haug [1] uggeted ung the the vrtual Euler parameter to repreent δξ uch that δ A Eδp (8) T ξ T where p e e e e1 e e3 the vector of Euler parameter ( e ) and E the Euler em-rotaton matrx defned a e1 e e3 e E e e e e 3 1! e e e e $ # 3 1 For example, p 3 the vector of Euler parameter aocated wth C 43. Multplyng Eq. (6) from the left by A yeld another property δπa ~ δa (1) Ung the relatonhp of vrtual dplacement and rotaton, Eq. (5) can be wrtten n term of the vrtual vector a δ ~ δ π Aδ (11) Subttutng the expreon for δ nto Eq. (3) yeld the varaton of the phercal contrant a δφ ph δr δr ~ δπ A δ ~ δπ Aδ (1) The contrant equaton wrtten n general form Φ( r, A,,C, r, A,C, ) (13) Notng that and C are varable, the varaton of Eq. (13) yeld δφ Φ δr Φ δπ Φ δr Φ δπ Φ δ Φ δ r π r π (9) 4

5 Φξ δξ Φξ ξ δ (14) Furthermore, the lnearzed contrant of Eq. (14) can be wrtten a δφφ δ z Φ δ z z z Φ δξ Φ δ (15) ξ where the extended vector of the vrtual ont-attachment vector δ δ! δ $ # (16) and the vrtual rotaton δξ δξ! δξ $ # (17) the coeffcent of vrtual dplacement are Φ z Φ r Φ π (18) Φ z Φ r Φ π (19) the coeffcent of vrtual ont-attachment and rotaton are Φ Φ Φ () Φ Φ Φ ξ ξ ξ (1) and the vrtual dplacement for bode and are δr δz! δπ $ # δr δz! δπ $ # () Note that the varaton of the contrant n Eq. (15) an extended expreon of the formulaton ued by Ta and Haug [], where the lat two term of the rght-hand de are ued to account for the varaton n ont-attachment and orentaton vector. IV. RECURSIVE FORMULATION In order to develop a formulaton for obtanng the varaton of the contrant functon Φ n term of the varaton of the generalzed ont coordnate ncludng the effect of degn varaton, Ta and Haug [] propoed wrtng the contrant varaton n the form of δφφ q δq (3) where δq the vector of relatve coordnate varaton and Φ q the Jacoban matrx n Jont pace. In th work, the ytem of equaton (Eq. 3) nclude the vrtual dplacement and rotaton vector a varable. A formulaton to recurvely obtan the tate varaton of body n term of the tate varaton of t uncton body and of the relatve coordnate along the chan wll be developed. The ytem of equaton (Eq. 3) may have more unknown than contrant. The Moore-Penroe peudo nvere then ued to obtan an aembled confguraton cloe to the ntal gue provded by a uer. The orthogonal tranformaton matrx reference frame can be wrtten a A from the body reference frame to the global 5

6 A ACA (4) where A the tranformaton matrx from the body reference frame xyz to the ont reference frame of body ( xyz ). Takng the varaton of Eq. (4) yeld δa δaca AδCA ACδA (5) Subttutng Eq. (6) and (1) nto Eq. (5) and multplyng by the vrtual rotaton matrxδ π ~ yeld δ~ δ~ ~ π π δξ δπ~ A ACA A CA AC A (6) where δ π can be obtaned from the relatve coordnate varaton δq, uch that δπ H 3A, q 8 δq (7) 3A, q 8 a tranformaton matrx that depend on the orentaton of body and the and H relatve coordnate q, whch defned for each type of ont. It can alo be hown that δπ can be wrtten a δπ δπ δξ Hδq (8) Referrng to Fg. 1, the orgn of body reference frame can be located by the poton vector gven by r r d (9) The varaton of Eq. (9) wa derved by Zou, et al. [3] and hown to be δr δr δ r r Aδ d d δ δ q q ~ ~ π( ) ξ (3) Addng a common term ( r δπ ) to both de of Eq. (3), and ung Eq. (6), Eq. (8), and the relatonhp ~ ~ ab ba yeld d δr ~ rδ δr ~ ~ π rδπ Aδ dδξ ~ rπ~ δq ~ rδξ (31) q Equaton (31) and the vrtual rotaton of Eq. (8) can be combned n matrx form a d δr rδ δr rδ A rh r d δ q δq δ! δ $ #! δ $# ~ π ~ π! $# ~ # ~ ~ ξ (3) π π H! I $ #! The tree graph of Fg. () repreent a par of lnk that are dconnected after cuttng a ont. Frt, we hall develop the general formulaton for uch a cae where a ont that connect bode and cut and that body p the uncton node of the two chan that contan bode and, repectvely. A chematc depcted n Fg. 3. $ # 6

7 -1 cut ont -1 m p n Bae Body Fg. 3. A par of bode dconnected after cuttng ont In th cae, the tate varaton of body cannot be repreented n term of the tate varaton of body nce no relatve coordnate ext between bode and. Therefore, the tate varaton of bode and mut be wrtten n term of the tate varaton of ther common node p and Eq. (15) can be wrtten a δφφ δ z $ Φ δ z $ z Φ δξ Φ δ $ (33) ξ where the tate vector can be wrtten a δz $ δz $ 1 Bδq M 1δ 1, Nδξ 1, (34) where B the velocty tranformaton matrx between bode and, defned by d( 1), ~ rh B q! H and the matrce M 1 and N are expreed a A 1 M 1 (36)! ~ ~ r d ( 1), and N! I $ # (37) Equaton (34) can be recurvely ued to obtan the tate varaton of body n term of the tate varaton of t common body p and of the relatve coordnate along the chan a δz $ δz $ B 1δq 1 M δ, 1 N 1δξ, 1 Bδq M 1δ, 1 Nδξ 1, (38) Smplfyng and wrtng n term of a ere yeld p k k k 1 k 1, k k k 1, k k m δz $ δz $ B δq M δ N δξ $# $ # (35) 7 (39) Smlarly, the tate varaton of body can be wrtten n term of a ere a 7

8 7 (4) p k k k 1 k 1, k k k 1, k k n δz $ δz $ B δq M δ N δξ Rewrtng the contrant functon n term of tate vector yeld!! 7 $#, 7 (41) ξ p k k k 1 k 1 k k k 1 k k m δφφz δz $ B δq M δ δξ $ N,, Φ $ z δz B q M ξ Φ Φ ξ $, N p kδ k k δ k k kδ k k δ δ k n $# Collectng mlar term and rearrangng yeld 7 B δ q M δ N δ δ k k k k k k k, k δ (4) ξ k k k 1 k 1 k k k 1 k k m δφφz $ B δq M δ, N δξ, Φ ξ Φ Φ ξ z$, k n where Φz Φ wa ued to elmnate the δ$z $ p term n Eq. (41). Expandng Eq. (4) and wrtng the ytem of equaton n the form of a Jacoban Φ q yeld δφφ q δq (43) where the Jacoban matrx n the ont pace and the vector of relatve coordnate varaton δq are defned by Φ Φ B... Φ B Φ M... Φ M Φ N... Φ N Φ B... and q z$ z$ z$ m m 1 1 m m Φ B Φ M... Φ M Φ N... Φ N Φ Φ $ $ $ $ $ n 1 1 n ξ (44) z z z z z δq δq... δq δ... δ δξ... δξ δq... δq m m 1, m 1, m 1, m 1, n δ n 1, n... δ 1, δξ n 1, n... δξ 1, δ δξ (45) Note that the number of varable wll, n general, be more than the number of contrant. Snce the Jacoban of Φ( q ) not quare, the problem of obtanng an aembled confguraton can be olved ung the Moore-Penroe peudo nvere (Allgower and Georg 199). Startng wth an ntal gue q 1, the new generalzed coordnate are calculated by evaluatng q Φ * ( Φ) (46) where Φ q * the Moore-Penroe peudo nvere of the Jacoban Φ q q T, defned by Φ q (47) Φ * T T q Φq ΦqΦq The new et of generalzed coordnate computed a q 1 q q. Th method converge to an aembled confguraton q wthn a few teraton (Allgower and Georg 199). V. EVALUATION OF KINEMATIC SENSITIVITY TERMS 8

9 Intal poton and orentaton etmate are pecfed. The ntal value of each ont attachment vector from ont reference frame xyz to body-fxed frame xyz are preented n Table x TABLE 1 Jont attachment vector n the body-fxed frame o o o o Etmate of generalzed coordnate are q T. For a gven ntal etmate that doe not atfy the contrant functon Φ, the Moore-Penroe teraton method ued to obtan an ntal aembly. The teratve algorthm contnued untl the ph magntude of all error and change n approxmate oluton atfy Φ k q ε e and q q ε, where ε e and ε are two mall number, k the number of contrant equaton, the number of generalzed coordnate, and the teraton number. The Jacoban matrx Φ ph q n ont coordnate pace tranformed from the Cartean pace ph ph ph ph Φq Φz B Φz B Φz B $ $ $ (48) The Jacoban matrx n the Cartean pace for the phercal ont, gven a ph Φ ~ ~ z $ I r 3 (49) and ph ph Φz Φ $ 3 (5) where B 1, B, and B 3 are velocty tranformaton matrce for the revolute ont, unveral ont, and revolute ont, repectvely, uch that B B r 1 g 1 B h 41 h 1 g 1 h 43 (51) where h 41 and h 43 are unt vector along the rotatonal ont axe n the global reference frame whch can be expreed by h 41 A 4 C 41 [ 1] (5) 1 C and h 43 A 4 C 43 1 y z 3 3 x y [ ] T (53) z 9

10 1 C where h 1 and g 1 are two unt vector along the rotatonal axe z 1 and y 1 n the global reference frame defned a h 1 A 1 C C g [ ] T (54) A C A ( q ) and the rotaton matrx A 11 wrtten a coq n q A 11 1q 6 nq coq 1! (56) and the varatonal generalzed coordnate are δq δq δq δq δq T Ung the Moore- Penroe method to update the generalzed coordnate q, only one teraton requred and the et o o o o of new generalzed coordnate are q T. After ntal aembly, the ont-attachment vector and rotaton vector ξ (or the Euler parameter vector p) are choen a degn parameter. In the patal four-bar mechanm, the degn parameter of ont-attachment vector n body-fxed reference frame are 41, 43, 1, 3 and 3. Once the confguraton change, the contrant equaton are volated, and there are three method to be performed untl contrant volaton are atfed to aemble the mechanm. (1) The generalzed coordnate q are changed uch that δq ph ph ph ph Φz B Φ Φ Φ B B $ 1δq1 q 3 3δ 4 δq T $ # (55)! $ # 3 (57) A degner, however, may chooe to mantan the generalzed coordnate a contant but elect to allow for a change n the dmenon of each body whch gve re to the followng method. () The ont attachment vector are changed uch that where ph ph ph ph ph ph Φz M Φ Φ Φ Φ Φ $ 4δ 41 z M $ 1δ 1 z M $ 4δ 43 δ3 δ3 (58) M 1! A1 $# M A 4 4! $# (59) ph Φ A (6) 3 1

11 ph Φ A 3 3 3) The generalzed coordnate q and ont-attachment vector are changed uch that ph ph δq ph ph ph ph Φz B Φ Φ Φ Φ Φ B M M B $ M 1δq1! 4δ 41 1δ 1 3 3δq4 3 4δ 43 δq3$# ph ph ph Φ δ Φ Φ 3 3 δ 3 3 (6) Smulaton reult due to changng 3 from [ 1. ] T to [ 1 ] T are preented n Table. Column preent the ntal etmate for a general confguraton. An ntal aembly computed and entered nto column 3. Ung Eq. (57), only the generalzed coordnate are allowed to change and thoe value computed for an aembled confguraton are hown n column 4. If only the ont-attachment vector are allowed to change ung Eq. (58), the computed aembly entered nto column 5 of Table. Fgure 4a depct a chematc of the mechanm after computng the new ont coordnate (method 1). Note that only ont angle have changed. Fgure 4b depct a chematc of the mechanm after computng only new ontattachment vector. Note that, even for a very mall tolerance (a norm of ~1 6 ), the method exhbt a hgh rate of convergence (wthn 4 teraton). TABLE Smulaton reult due to changng varable ntal etmate ntal aembly q q, q q q q P P [ -1. ] T [ -1 ] T [ -1 ] T [ -1 ] T (61) 11

12 P Φ norm 1.547e e e e-7 Iteraton number B 3 B 3 q B q B 3 A 3 A 3 A 3 A 3 q A q A 3 q B 4 q A 1 q B 1 q A 4 Fg. 4 Aembled mechanm ung (a) method 1 and (b) method If the length of lnk, however, reduced from 1. to 6, t geometrcally mpoble that an aembled confguraton be obtaned nce lnk and lnk 3 cannot be connected by a phercal ont. Neverthele, an aembled confguraton can be obtaned ung the other two method. In fact, t ha been our experence that method () and (3) are more practcal from a degn pont of vew. Another poblty are for whch the Euler parameter can be changed. In uch a cae, the orentaton of a body can be altered,.e., varyng p 1, p and p 3. There are three practcal method to be performed. 1) The generalzed coordnate q are changed (Eq. 57). ) The generalzed coordnate q and ont attachment vector are changed (Eq. 58). 3) The generalzed coordnate q, ont attachment vector, and Euler parameter p are changed uch that ph ph δq ph ph ph ph Φz B Φ Φ Φ Φ ( ) Φ ( ) B M M N A E p N A E p $ 1δq1! 4δ 41 1δ δ 1 1 δ δq3$# Φ ph z B Φ Φ ( ) Φ Φ Φ M N A E p $ 3 3δq4 ph 3 4δ 43 ph 3 3 ph ph ph 4 3 δ 3 δ 3 3 δ 3 3 (63) where ~ r1 ~ r N1! I $ # ~ r3, N, N3 I I! $#! $# 1

13 Euler parameter p 3 are changed from [ ] T to [ ] T, thu the correpondng tranformaton matrx C 43 from revolute ont reference frame to ground reference frame changed from to 1 Ung the three method decrbed above, the ytem can be aembled atfactorly. The reult are preented n Table 3. Intal etmate are entered nto column. An ntal aembly computed and entered nto column 3. For a change n the lnk orentaton whle keepng contant ont-attachment vector, the generalzed coordnate are computed and entered n column 4. Note that 34 teraton were requred to compute q. If the generalzed coordnate and the ontattachment vector are computed ung Eq. (58), only 1 teraton are needed to converge to an aembled confguraton the reult of whch are entered nto column 5. If the generalzed varable, the ont-attachment vector, and the Euler parameter are computed ung Eq. (63), only 8 teraton are needed to converge to a oluton. The reult are entered nto column 6. TABLE 3 Smulaton reult due to changng p 3 1 varable ntal etmate 3 ntal aembly 4 q 5 6 q, q, and p q q q q [ -1. ] T P P

14 P Φ norm 1.547e e e e-6 Iteraton number It can be notced, n th cae a n mot cae that have been mplemented ung th expermental code, that allowng the code to compute generalzed coordnate, ont attachmentvector, and Euler parameter (method 3), exhbt the hghet rate of convergence. VI. CONCLUSIONS An analytcal formulaton for tudyng the knematc entvty of the patal four-bar mechanm preented. The formulaton preented n th paper and llutrated by a numercal example demontrate the valdty of a general purpoe formulaton and expermental computer code for the computaton of degn parameter after an alteraton n the orgnal degn. The patal four-bar mechanm modeled ung graph theory and cloed loop are converted to a tree tructure ung a cut-ont contrant formulaton. It wa hown that th formulaton can be derved wth repect to degn parameter to obtan knematc entvty. Thee contrant are derved keepng the ont-attachment vector and orentaton matrce (Euler parameter) a degn varable. Varaton of thee vector and matrce were alo developed. It wa hown that the Jacoban matrx n Cartean pace can be tranformed to ont coordnate pace to compute varaton. It wa hown that tartng from an ntal confguraton that atfe the contrant, applyng a degn change to the ont-attachment vector or Euler parameter, can be propagated throughout the mechanm. It wa hown that an aembled confguraton can be obtaned ung the Moore-Penroe peudo nvere method. The method ha howed to be applcable for a varety of degn parameter. It ha been hown that the method of changng lnkage length whle mantanng contant generalzed coordnate the fatet method to obtan a new et of varable. Th method ha alo exhbted the hghet accuracy. The knematc entvty due to changng the orentaton of a body ha howed a hgher rate of convergence to an aembled confguraton ung the method of allowng the generalzed coordnate, ont-attachment vector, and Euler parameter to be automatcally computed. It ha been our experence that allowng the maxmum number of degn parameter to change, reult n hgher potental for convergence. ACKNOWLEDGMENTS Th reearch wa funded by the US Army Tank Automotve Command Center (TACOM) through the Automotve Reearch Center (Department of Defene contract number DAAE

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16 3. H. L. Zou, K. Abdel-Malek, and J. Y. Wang, ASME Journal of Mechancal Degn (ubmtted). 16